Summary Table

Key parameters for each production dataset have been summarized and compiled into a table. Cumulative months on production, initial oil and gas rates, peak oil and gas rates, peak month, and cumulative volumes can be seen for each UWI. These parameters will be used in subsequent calculations and help us prioritize wells as we begin to look at learning from wells with more production history later in PDCA process.

Production Plots

Modified Hyperbolic

Explanation of that equation and a definition using code..?

We’ll define the modified hyperbolic equation in executable form. The equation will accept decline parameters and a time series across which it can evaluate production rate. First we’ll need a function to convert secant decline percentage into nominal form.

## to.nominal function converts effective decline rates to nominal decline rates
to.nominal<-function(De,b=0){
  if(b==0){
    a <- -log(1-De)
  } else if(b==1){
    a <- De/(1-De)
  } else{
    a <- (1/b) * (((1-De)^(-b))-1)
  }
  return(a)
}

##  MH function is the modified hyperbolic function, where a single b value transitions 
##  to an exponential decline at a limiting decline rate
# Function will accept equation constants and an elapsed time value in days?
# Output will be rate at given time for given parameters
MH<-function(qi, b = 1.5, Di = 0.7, Dlim = 0.07, q.abd = 3, t = c("Enter In Days")){
  #qi must be entered as a rate
  #Di must be decimal in effective secant format as % year
  #t must be entered as days
  
  #Convert Di and Dlim to nominal
  ai <- to.nominal(Di)     
  alim <- to.nominal(Dlim)  
  
  #Convert t to yearly time fraction
  t <- t/365.25
  
  #Add error handlers for true exponential decline and for if intial parameters
  # bypass the exponential decline function.. I.e. if di<dlim.. error
  #If di > dlim, use Di and no Dlim until end of time
  if(b==0){
    #Exponential Decline
    q<-qi*exp(-ai*t)
  } else{
    #Calculate limiting decline rate
    qlim <- qi*(alim/ai)^(1/b)
    
    #Calculate limiting decline time, in years
    tlim <- (((qi/qlim)^b)-1)/(b*ai)
    
    #Calculate q assuming hyperbolic
    q<-qi/((1+b*ai*t)^(1/b))
    
    #Replace q when t > tlim
    q[t > tlim] <- qlim*exp(-alim*(t[t>tlim] - tlim))
    
    #Abandonment cutoff. When q < q.abd, q = 0
    q[q < q.abd] <- 0
    
  }
  return(q)
}

Combine Sampled Parameters

Let’s combine and organize our sampled parameters in a data frame. We’ll use an equation-specific function to ensure completeness and repeatability for multi-well matching.

# The MHSample function will sample and combine all parameters based on their 
# DefineDistribution classes. Method input will return different output type based on 
# requirement to speed up MCMC calculations in the future.. Ignore for now.?
MHSample<-function(n=10000,qi,b,di,dlim){
  #Sample relevant parameters for MH function: qi, b, di, dlim
  #Inputs of class distribution created prior to function call, containing distribution information
  # for each parameter
  #Function will output a Monte Carlo matrix with n x m dimensions, where n is sample number
  # and m is the number of parameters needed for MH equation
  
  #Step through each parameter and call sample.any function
  qi.out <- sample.any(n,qi)
  b.out <- sample.any(n,b)
  di.out <- sample.any(n,di)
  dlim.out <- sample.any(n,dlim)
  
  #Combine each returned column into dataframe for output
  sampledParams <- data.frame("qi"=qi.out,"b"=b.out,"di"=di.out,"dlim"=dlim.out)
  
  #Return sampled dataframe
  return(sampledParams)
}

# Pass the previously defined distributions into the MH.sample function
# Note this is repeating our earlier sampling in a cleaner form
MH.sampled.parameters <- MHSample(n = 10000,qi = qi,b = b,di = di,dlim = dlim)

# Return in table, displaying first 10 rows only
kable(MH.sampled.parameters[c(1:10),], align = 'c') %>% 
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive",
                                      full_width = F, position = "center"))

Historical Rate Synthesis

Using the combined parameter dataframe and the modified hyperbolic equation, we can define a function to use each row of the parameter dataframe as a Monte Carlo iteration and substitute it into the arps equation across the time interval of the current well. The function will create and return a mcIteration object for each iteration. This object is a list with slots for Iteration name, Parameters, and Synth. Some additional slots will be created and populated later as we approach cost and forecasting.

# A function, MH.synthesize, will be defined to carry out rate synthesis across the actuals 
# time interval.
MH.synthesize<-function(parameters,q.abd,t){
  #Parse up parameters and pass into MH function call
  fc <-  MH(parameters[["qi"]],parameters[["b"]],parameters[["di"]],parameters[["dlim"]],q.abd,t)
  
  #Simplify parameters into a single string representation
  parametersName<-paste(names(parameters),round(parameters,digits=2),sep = "=",collapse=" ")
  
  #Create mcIteration imitation list to improve performance and ease of combining data
  #Return mcIteration. Remaining list items to be appended by later functions
  return(list("Iteration" = parametersName,"Parameters" = parameters,"Synth"=fc))
}

# Define a time interval of actuals to synthesize across, using 100/05-17-081-16W6/00
uwi<-"100/05-17-081-16W6/00"

# Subset production including only 100/05-17-081-16W6/00
singleWell.all <- formated_production$CD[which(formated_production$CD$Unique.Well.ID == 
                                              uwi),]

# Find peak oil month for this uwi
uwi.pm <- matrix[["Oil.Peak.Month"]][which(matrix$eachUWI==uwi)]

# Refine production datatable to only include data after peak oil rate
singleWell<-singleWell.all[-c(1:(uwi.pm-1)),]

# Create time vector starting at 0 for peak oil month
synth.time <- (singleWell$CD.Month - uwi.pm) * 30.4 # Converted to daily

# Let's apply the MH.synthesize function in an apply call to execute across each row
# of the parameters dataframe
oil.synth<-apply(MH.sampled.parameters,1,MH.synthesize,q.abd = o.abd, t = synth.time)

# Print first object of oil.synth to show mcIteration object
print(oil.synth[[1]])

## $Iteration
## [1] "qi=566.08 b=1.89 di=0.47 dlim=0.07"
## 
## $Parameters
##         qi          b         di       dlim 
## 566.077933   1.885302   0.473363   0.070000 
## 
## $Synth
##  [1] 566.0779 538.0106 513.6163 492.1628 473.1066 456.0348 440.6274 426.6323
##  [9] 413.8474 402.1089 391.2825 381.2565 371.9376 363.2471 355.1178 347.4921
## [17] 340.3206 333.5602 327.1732

Each mcIteration list, defining a Monte Carlo iteration and historical synthesis, can now be evaluated against the actual data to search for an appropriate model.

Soft L1

Soft L1 loss is a robust, continuous variation of the common Huber Loss that gives less weight to outliers than a L2 loss, making it useful for fitting time series production data. Our Soft L1 definition will return the individual loss of each point and an average cost for all residuals input.

soft_L1<-function(r){
  IndvCost <- 2*((1 + abs(r))^0.5 - 1)
  Cost <- sum(IndvCost,na.rm = TRUE) / length(IndvCost)
  return(list(IndvCost,Cost))
}

To apply the loss equation we’ll define a function to apply the Soft L1 loss to the residuals of each model match. The function will include an option to apply the loss function against the log residuals, an option that we’ll take advantage of due to the non-linear nature of production data. The function will populate the IndvCost and Cost slots of each mcIteration object, and once the function is defined, we can use these slots to visualize each model’s fit.

## applyLoss function will be used to accept a list of mcIteration objects, an actuals dataframe,
# a loss function to apply, and an option to use log residuals or absolute residuals, 
# then populate the cost and avg_cost slots of the mcIteration objects and return them.
# Bestfit will contain c(eps_mean_min,eps_std_min) for pdca

applyLoss<-function(iterationObject = c("mcIteration List"),
                    actuals = c("Matrix with actual data"), loss.function = soft_L1,
                    log.residuals = TRUE){
  
  # Calculate residuals from model synthesis to actuals. If log.residuals is TRUE, 
  # return the log of the residuals
  if (log.residuals == TRUE){
    r <- log(actuals) - log(iterationObject$Synth)
  } else {
    r <- actuals - iterationObject$Synth
  }
  
  # Apply loss function to residuals. Returns a list of IndvCost and Cost.
  rho <- loss.function(r)
  
  # Append IndvCost and avg_cost (named cost) to the mIteration imitation list calling from
  # rho list slots
  iterationObject$IndvCost <- rho[[1]]
  iterationObject$Cost <- rho[[2]]
  
  # return iterationObject
  return(iterationObject)
}

# Use the applyLoss function in an lapply call across each object in oil.synth
oil.synth <- lapply(oil.synth, applyLoss, actuals = singleWell[["Oil.Rate"]], 
                    loss.function = soft_L1, log.residuals = FALSE)

# Repeat for log residuals
oil.synth.log <- lapply(oil.synth, applyLoss, actuals = singleWell[["Oil.Rate"]], 
                    loss.function = soft_L1, log.residuals = TRUE)

# Updated mcIteration object
print(oil.synth[[1]])

## $Iteration
## [1] "qi=566.08 b=1.89 di=0.47 dlim=0.07"
## 
## $Parameters
##         qi          b         di       dlim 
## 566.077933   1.885302   0.473363   0.070000 
## 
## $Synth
##  [1] 566.0779 538.0106 513.6163 492.1628 473.1066 456.0348 440.6274 426.6323
##  [9] 413.8474 402.1089 391.2825 381.2565 371.9376 363.2471 355.1178 347.4921
## [17] 340.3206 333.5602 327.1732
## 
## $IndvCost
##  [1] 28.88406 32.56231 34.73612 35.90350 35.83259 35.68141 34.42292 34.88264
##  [9] 34.77318 34.58031 34.41093 34.43722 33.83672 33.68369 33.30451 32.98297
## [17] 32.87979 32.56150 32.27778
## 
## $Cost
## [1] 33.82285

With cost calculated using the Soft L1 loss function applied to the standard and log residuals for every model iteration, let’s use the average cost to find our “best fit” model out of the 10,000 iterations. We’ll plot the model against the actual oil rate, coloured by individual cost for both residual method to look at how each handles outliers.

# Use lapply and do.call to combine the average cost of every model object 
# into a single dataframe
oil.cost <- do.call(rbind,lapply(oil.synth,function(x){
  return(x$Cost)
}))

# Repeat for log residuals
oil.cost.log <- do.call(rbind,lapply(oil.synth.log,function(x){
  return(x$Cost)
}))

# Find index of best fit model using log soft_L1 residuals
oil.bf.log <- which.min(oil.cost.log)

# Plot this "best fit" against actuals and colour by standard and log residual cost
# To plot, we'll utilize some custom plotly functions defined in plotResources external file
oil.synth.plot <- plotly.multimodel.indvcostcolor(actuals.x = singleWell[["CD.Month"]],
                                                  actuals.y = singleWell[["Oil.Rate"]],
                                                  iterationObject = oil.synth[oil.bf.log],
                                                  scales = "semi-log")
oil.synth.plot <- layout(oil.synth.plot, xaxis = list(title = "Cal-Day Month"),
                         yaxis = list(title = "Oil Rate (bbls/d)"), 
                         title = "Soft L1 Cost of Residuals")

oil.synth.plot.log <- plotly.multimodel.indvcostcolor(actuals.x = singleWell[["CD.Month"]],
                                                  actuals.y = singleWell[["Oil.Rate"]],
                                                  iterationObject = oil.synth.log[oil.bf.log],
                                                  scales = "semi-log")

oil.synth.plot.log <- layout(oil.synth.plot.log, xaxis = list(title = "Cal-Day Month"),
                         yaxis = list(title = "Oil Rate (bbls/d)"), 
                         title = "Soft L1 Cost of Log Residuals")
# Combine into a subplot
combined<-subplot(oil_prod,gas_prod,nrows = 2,shareY = TRUE,titleX = FALSE,titleY = TRUE)
combined<-layout(combined,showlegend=FALSE,title="Oil & Gas Production",xaxis2=list(title="Cal-Day Month"))

combined.cost <- subplot(oil.synth.plot,oil.synth.plot.log,nrows = 1,titleX = TRUE,titleY=FALSE)
combined.cost <- layout(combined.cost,title = "Soft L1 Cost of Absolute vs. Log Residuals",
                        yaxis=list(title="Oil Rate (bbls/d)"))
#IMPROVE TOOLTIP???
hide_colorbar(combined.cost)

Comparing the absolute residual cost in the left plot against the log residual cost in the right plot, there is similar relative spread in the residuals and both cost the peak rate similarily, but the log residuals does a better job at costing the sub peak at month 9. We will carry log residual cost through the remainder of the CAST methology as it capture the non-linearity of the production data better. One important thing to notice before moving on is that the Soft L1 function gives a high cost to the peak oil rate and does not match it perfectly as a result. This may be concerning when compared to traditional DCA matches, but given that early time production data can be far from steady-state flowing behaviour, it is a useful feature to maintain.

Using the log residuals and the Soft L1 cost function, let’s find the top 100 models from our initial 10,000 iterations and see how they describe the ranges of our actual oil rates. We’ll colour them by average cost to compare the forecasts.

# Find and plot top 100 (top 1%) models using the ggplot resource. Colour by cost.
oil.synth.P1 <- which(oil.cost.log <= quantile(oil.cost.log,0.01))

#Sort from highest cost to lowest for plotting visuals
oil.synth.P1 <- oil.synth.P1[order(oil.cost.log[oil.synth.P1],decreasing = TRUE)]


oil.plot.P1 <- ggplot.multimodel.costcolor(actuals.x = singleWell[["CD.Month"]],
                                           actuals.y = singleWell[["Oil.Rate"]],
                                           iterationObject = oil.synth.log[oil.synth.P1],
                                           scales = "semi-log")
# Add titles and output
oil.plot.P1 <- oil.plot.P1 + 
  xlab("Cal-Day Month") +
  ylab("Oil Rate (bbls/d)")

oil.plot.P1

Top 1% of Models Using Soft L1 Loss

We can see that the top 100 forecasts do a good job of covering the range of actual oil rates, with the lowest cost forecasts tightly grouped around the actuals. We’ve shown we can use Monte Carlo sampling paired with a loss function to match historical production! Before moving on let’s note that the majority of the forecasts are under-predicting the intial rate. This is primarily driven by the high cost of the peak oil rate using the Soft L1, but could also be improved through our input distributions. This is something we can improve upon later.

PDCA Loss

Explain this loss function, where it’s from, and show a comparison of application to the Soft L1 loss

The Soft L1 loss function does a decent job at finding a best fit model, but is there a better loss function that can be used? Let’s look at a more complete loss function defined in SPE-174784-PA (ADD LINK HERE AS REFERENCE) by Fulford et al. takes a similar approach, but .. EXPLAIN HERE

# As our Soft L1 was defined, pdca_loss will slots for IndvCost and Cost
pdca_loss<-function(r, Eps_mean_min = 0, Eps_std_min = 0){
  IndvCost <- r^2 # pdca loss simply uses L2 loss for individual cost
  
  Eps_mean <- mean(r,na.rm = TRUE) #Mean of residuals
  Eps_std <- sd(r,na.rm=TRUE) #Std of residuals
  
  Cost <- ((Eps_mean - Eps_mean_min)/.1)^2+((Eps_std - Eps_std_min)/.01)^2
  return(list(IndvCost,Cost))
}

Just as we did with the Soft L1, let’s use the PDCA loss function to find the top 100 forecasts from the same 10,000 iterations. We’ll see the minimum mean and minimum standard deviation of epsilon to 0, assuming a perfect model. Again we’ll colour by average cost and compare to the Soft L1 method.

# Let's modify the applyLoss function slightly to accept the pdca bestfit parameters
# eps_min input will contain c(eps_mean_min,eps_std_min) for pdcaLoss
applyLoss<-function(iterationObject = c("mcIteration List"),
                    actuals = c("Matrix with actual data"), loss.function = soft_L1,
                    log.residuals = TRUE,
                    eps_min = c(0,0)){
  
  # Calculate residuals from model synthesis to actuals. If log.residuals is TRUE, 
  # return the log of the residuals
  if (log.residuals == TRUE){
    r <- log(actuals) - log(iterationObject$Synth)
  } else {
    r <- actuals - iterationObject$Synth
  }
  
  # Apply loss function to residuals. Returns a list of IndvCost and Cost.
  # Since pdca_loss is used, pass bestfit. Default to 0,0 as ideal bestfit.
  rho <- loss.function(r,eps_min[1],eps_min[2])
  
  # Append IndvCost and avg_cost (named cost) to the mIteration imitation list calling from
  # rho list slots
  iterationObject$IndvCost <- rho[[1]]
  iterationObject$Cost <- rho[[2]]
  
  # return iterationObject
  return(iterationObject)
}

# Apply PDCA loss function
oil.synth.PDCA <- lapply(oil.synth, applyLoss, actuals = singleWell[["Oil.Rate"]], 
                    loss.function = pdca_loss, log.residuals = TRUE)

# Use lapply and do.call to combine each object loss into a single dataframe
oil.cost.PDCA <- do.call(rbind,lapply(oil.synth.PDCA,function(x){
  return(x$Cost)
}))

# Find and plot top 100 (top 1%) models using the ggplot resource. Colour by PDCA cost.
oil.synth.pdcaP1 <- which(oil.cost.PDCA <= quantile(oil.cost.PDCA,0.01,na.rm = TRUE))

#Sort from highest cost to lowest for plotting visuals
oil.synth.pdcaP1 <- oil.synth.pdcaP1[order(oil.cost.PDCA[oil.synth.pdcaP1],decreasing = TRUE)]


oil.plot.pdcaP1 <- ggplot.multimodel.costcolor(actuals.x = singleWell[["CD.Month"]],
                                           actuals.y = singleWell[["Oil.Rate"]],
                                           iterationObject = oil.synth.PDCA[oil.synth.pdcaP1],
                                           scales = "semi-log")
# Add titles and output
oil.plot.pdcaP1 <- oil.plot.pdcaP1 + ggtitle("PDCA Loss Function") +
  xlab("Cal-Day Month") +
  ylab("Oil Rate (bbls/d)")

oil.plot.P1 + ggtitle("Soft L1 Loss Function")
oil.plot.pdcaP1

Comparing Top 1% of Models Using Soft L1 and PDCA Loss

Before we move on to forecasting, let’s look at the parameter distributions for the top 1% of models to get an idea of the solution space using the pdca loss method.

# Combine matched parameters into single data frame and plot using three histograms (1 for qi,
# 1 for b, and 1 for di). This will give us a good representation of the solution space.

# Combine P1 parameters into a dataframe
acceptedParameters <- do.call(rbind,lapply(oil.synth.PDCA[oil.synth.pdcaP1],function(x){
  return(cbind(x$Parameters[1],x$Parameters[2],x$Parameters[3]))
}))

colnames(acceptedParameters)<-c("qi","b","di")
rownames(acceptedParameters)<-c()
acceptedParameters<-as.data.frame(acceptedParameters)
# Display distributions using ggplot2 histograms
ggplot(data = acceptedParameters, aes(x = b)) +
  geom_histogram(aes(y= ..density..), colour = "black", fill = "grey", alpha = .5) +
  geom_density(alpha=.2, fill="#763F98") +
  theme_classic() +
  theme(text = element_text(size = rel(5)))

ggplot(data = acceptedParameters, aes(x = di)) +
  geom_histogram(aes(y= ..density..), colour = "black", fill = "grey",alpha=.5) +
  geom_density(alpha=.2, fill="#763F98") +
  theme_classic() +
  theme(text = element_text(size = rel(5)))

ggplot(data = acceptedParameters, aes(x = qi)) +
  geom_histogram(aes(y= ..density..), colour = "black", fill = "grey",alpha=.5) +
  geom_density(alpha=.2, fill="#763F98") +
  theme_classic() +
  theme(text = element_text(size = rel(5)))

Parameters of P1 model matches using PDCA Loss method

Summarize observations from these distributions.. They are mostly lognormal. Therefore, do we need to step into MCMC to get a proper PDCA distribution going forward? Keep these in mind and compare later on..

Summarize findings in these two charts and how we’re going to carry PDCA loss going forward in the rest of the evaluation / process as it provides a better probabilistic representation. Show distribution fits of the accepted parameters? Or do this at a later time? Or here just FYI??

Burn-In

Use original MCMC run as burn-in.

Jumping function and distributions

To proceed through the Metropolis-Hasting algorithm for MCMC, we need to define jumping distributions and use them to propose the next test parameter set.

# Define mcmc.eval function to evaluate sampled jump for acceptance probability. 
# Returns x_i+1 if accepted, else, xi
# eps_min input used to pass eps_mean_min and eps_std_min of bestfit for pdca_loss function

mcmc.eval <- function(currentIteration = c("mcIteration object defining x_i"),
                          sampledParms = c("dataframe of next iteration parameters"),
                          loss_function = c("loss function to evaluate acceptance"),
                          synth_function = c("synthesis function"),
                          actuals = c("data frame with actuals"),
                          actual.t = c("time to synthesize"),
                          q.abd = c("abandonment rate"),
                          eps_min = c(0,0)){
  
  # Synthesize data using sampledParms and synth_function
  proposed <- synth_function(sampledParms,q.abd,actual.t)
  
  # Call applyLoss function to evaluation sampled parameters
  proposed <- applyLoss(proposed,actuals,loss_function,log.residuals = TRUE,eps_min)
  
  # Evaluate acceptance probability
  A <- exp(-1/2*proposed$Cost) / exp(-1/2*currentIteration$Cost)
  
  # Evaluate acceptance
  if(runif(1)<A){
    output <- proposed
  } else{
    output <- currentIteration
  }
  
  # Return proposed mcIteration object and acceptance probability
  return(output)
}

# Define MH.mcmc to perform MCMC Metropolis Hastings algorithm for modified hyperbolic model
# Returns list "chain" of accepted mcIteration objects.


MH.mcmc <- function(iterations = 10000,
                    initial = c("starting mcIteration object"),
                    loss_function = c("loss function to evaluate acceptance"), 
                    jump = c("list of jumping distributions"), 
                    bounds = c("list of bounding distributions"),
                    actuals = c("actual rates"), 
                    actual.t = c("actual time"), q.abd = c("abandonment rate")){
  # Initialize chain
  chain <- vector("list",iterations)
  
  # Assign initial mcIteration object to chain
  chain[[1]] <- initial
  
  # Calculate eps_mean_min and eps_std_min from initial as bestfit for pdca loss
  r <- log(actuals) - log(initial$Synth)
  eps_min <- c(mean(r,na.rm = TRUE),sd(r,na.rm = TRUE))
  
  # Define bounds for each parameter
  bound <- lapply(bounds,function(x){
    if(x@type == "Uniform"){
      c(x@min_mean,x@max_sd)
    } else if(x@type == "Normal"){
      c(qnorm(0.01,x@min_mean,x@max_sd),qnorm(0.99,x@min_mean,x@max_sd))
    } else {
      c(qlnorm(0.01,x@min_mean,x@max_sd),qlnorm(0.99,x@min_mean,x@max_sd))
    }
  })
  
  # Perform MCMC Metropolis Hasting algorithm across number of iterations
  for (i in c(2:iterations)){
    # Assign starting points of jumping distributions to previous iteration
    # Use ln values for qi and di
    jump[[1]]@min_mean <- log(chain[[i-1]]$Parameters[[1]])
    jump[[2]]@min_mean <- chain[[i-1]]$Parameters[[2]]
    jump[[3]]@min_mean <- log(chain[[i-1]]$Parameters[[3]])
    jump[[4]]@min_mean <- chain[[i-1]]$Parameters[[4]]
    
    # Sample model parameters using jumping distributions
    sampledParms <- apply(MHSample(1,jump[[1]],jump[[2]],jump[[3]],jump[[4]]),2,
                          function(x){x})
    
    # Revert log parameters
    sampledParms[c(1,3)]<-exp(sampledParms[c(1,3)])
    
    # Compare sampled parameters to bounding distributions
    if(sampledParms[["qi"]] <= bound[[1]][2] & sampledParms[["qi"]] >= bound[[1]][1] &
       sampledParms[["b"]] <= bound[[2]][2] & sampledParms[["b"]] >= bound[[2]][1] &
       sampledParms[["di"]] <= bound[[3]][2] & sampledParms[["di"]] >= bound[[3]][1] &
       sampledParms[["dlim"]] <= bound[[4]][2] & sampledParms[["dlim"]] >= bound[[4]][1]){
      # Accept and call mcmc.evel to evaluate jumping parameters and append to chain
      # mcmc.eval will return x_i+1 if accepted or x_i if not
      chain[[i]] <- mcmc.eval(chain[[i-1]],sampledParms,loss_function,MH.synthesize,
                            actuals,actual.t,q.abd,eps_min)
    } else{
         # Reject parameters and keep previous iteration
      chain[[i]] <- chain[[i-1]]
       }
  }
  
  # return chain with MCMC samples
  return(chain)
}

# Using best fit from initial monte carlo calls, perform metropolis hasting algoritm
# Define best fit

# Combine costs from oil.forecast
# Use lapply and do.call to combine each object loss into a single dataframe
bestfit.cost <- do.call(rbind,lapply(oil.forecast,function(x){
  return(x$Cost)
}))

# Find min cost as best fit
bestfit <- oil.forecast[[which.min(bestfit.cost)]]

# Define jumping distributions as DefineDistribution objects
qi_jump <- new("DefineDistribution",type = "Normal",min_mean = 0, max_sd = 0.2)
b_jump <- new("DefineDistribution",type = "Normal",min_mean = 0, max_sd = 0.2)
di_jump <- new("DefineDistribution",type = "Normal",min_mean = 0, max_sd = 0.4)
dlim_jump <- new("DefineDistribution",type = "Uniform",min_mean = o.dlim, max_sd = o.dlim)

# Combine into single list
jump <- list(qi_jump,b_jump,di_jump,dlim_jump)

# Combine bounding distributions into bound
bounds <- list(qi,b,di,dlim)

# Call MH.mcmc to perform MCMC evaluation
oil.mcmc <- MH.mcmc(iterations = 10000, initial = bestfit,
                    loss_function = pdca_loss, jump = jump, bounds = bounds, 
                    actuals = singleWell[["Oil.Rate"]], actual.t = synth.time, q.abd = o.abd)

# Need any sort of "kick away from these bounds? Or let it iterate until it gets away itself?
# Look at changing the jumping distributions as well

# Add Di !< 1 error handler in MH function as well?

oil.mcmc.plot <- ggplot.multimodel.costcolor(actuals.x = singleWell[["CD.Month"]],
                                           actuals.y = singleWell[["Oil.Rate"]],
                                           iterationObject = oil.mcmc,
                                           scales = "semi-log")
# Add titles and output
oil.mcmc.plot <- oil.mcmc.plot + 
  xlab("Cal-Day Month") +
  ylab("Oil Rate (bbls/d)") +
  ggtitle("PDCA Loss MCMC Output")

# Plot Monte Carlo PDCA plot
oil.plot.pdcaP1

# Add mcmc plot to output
oil.mcmc.plot

Comparing Top 10% Monte Carlo Matches using PDCA Loss to MCMC PDCA Loss Output

We can see that although the MCMC output may require some tuning, it returns 10,000 matches that describe the actuals data very well in a probabilistic manner. They have a tighter spread around the actuals with more forecasts for probabilistics. Let’s look at the distributions of the matched parameters to look for improvements over the top 100 matches from the initial Monte Carlo run.

Accepted match parameters using MCMC method and PDCA Loss

The resulting distributions are dependent on the jumping distributions, which need to be tuned appropriately, but the accepted distributions follow both normal and lognormal distributions. How do the probabilistic forecast outcomes look for this MCMC run?

Results